You're not alone. I'm a paediatrician and a manager, and a few years ago, found myself in your position - every minute of nearly every day was filled with either patient appointments or meetings (I was the medical director of my organisation at the time).
One day I decided I'd had enough of never being able to get anything done and made a list of all my fixed commitments. If I went to every meeting and clinic I was supposed to attend it added up to 44 hours a week, and they only pay me for 40! I also made a list of all the major projects I was supposed to be working on for the organisation.
So then I made an appointment with the Chief Executive, who I reported to, and took all this information to her. Fortunately for me, she got the message quickly, and we renegotiated some of the stuff so I got a weekly four hour session which was NOT hard landscape.
Since then I've had to do several more renegotiations as my job has changed (I'm not medical director any more). My current schedule has about 4 hours a week, if I'm lucky, for "doing". This week, I didn't have any, but managed to pinch an hour from a quiet clinic. As an emergency measure I have brought some work home for the weekend, but I don't usually do that more than once every 2 months or so.
From personal experience I'd say that if you don't have at least 4 hours a week with time for doing, then you are set up to fail, because it's not physically possible to do all the the non-meeting things if you don't have the time.
It might be worthkeeping a time diary for a few weeks and assessing just how much time you really need for this job, before going to talk to your boss.
and good luck!
Ruth